I sometimes see attempts at deriving the born rule, 'or what i call the quantum principal principle' which make an empirical assumption (no signalling) to justify what appears to an invariance/'rationality or indifference assumption' This sometimes appears, both in the everett/non-gleason-like, envariance proofs, and in the more conventional gleason style proofs.
I might be wrong or muddleheaded in saying this but seems back to front or irrelevant ( a category error maybe). Even if there is 'no signalling' between two of the contexts, or between the pairs in an actual entangled systems (' both actual existent but numerically distinct lhs and rhs entangled particles') , or pairs or tuples theirof; Because even if there is 'no signalling or causation' between the two systems, or even if no ' mere non causal' strict anti-correlation' between the two systems, this seems is beside the point.
That is unless one can justify the two systems being strictly correlated with regards to the truth values of the same measurement outcomes , (and not just their supposed probabilities).
Even so, in some cases, the non signalling would also tell against this being the case (strict positive deterministic correlation) being the case. And in so doing, well thus undermining a (justifiable inference of rationality) based on necessarily equal truth values,
That is unless 'no-signalling' is used to justify a 'perfect strict correlation of truth values'. I presume this is is not what is meant by correlated twin observables/unitaries or the perfect correlation principal (of barnum 2008), because different contexts can have different outcome measurement/ values, or so it is argued within the KS theorem.I).
And generally these differences are needed, at the next stage of the argument,TO GIVE a justified necessarily 'equal truth value, equal probability value' inference for the opposite outcome in each context at the next stage of the argument/proof.
That is, because the fact that the other system necessarily has an opposite measurement outcome truth value (via some complex entanglement/contextuality), this is then used at the next part of the argument (acceptably now) to justify that this opposite outcome (spin down say) in context 2, has an equal probability value to the probability of the original outcome (spin up) in context/particle 1.
This being due to the 'necessarily equal truth value, equal probability principle' which appears to be acceptable non-probabilistic or 'non circular' rationality principal in this regard; this being opposed to a 'mere epstemic/ objective symmetry/ logical symmetry'in each distinct context/particle, to get an equal truth value, equal probability link, /justified which appears somewhat circular
Presumably one is not attempting to justify (by no signalling) that the 'amplitude norm squared is equal' in both systems/ contexts, that is a given. (ie likewise one could say that the chances/propensities, subjective credences, relative frequencies, or whatever you, or that the value of 'probability' under 'one's contingent/physical chance/probability interpretations are equal; 'that is, what features in the rhs of the born rule/ principal principle, to be a given' because that is not what one is attempting to justify. I call the above entities 'norm amplitudes/normed squared amplitude interpretations/chances' just (A) probability interpretations
Given (A) are equal, presumably one is attempting to justify that (B) the 'logical/metalinguistic probabilities', or 'rational objective degrees of belief' in both/all contexts are equal, ( given already that the same amplitudes (norm squared) values/chances (A) are equal in that other context). These are what feature in the LHS of the PP or born Rule, that is the thing one is attempting to prove.
Therefore to say that because the amplitudes norm squared or the merely the chances must be equal in each context due to non-signalling is not enough to get that that the (B) probabilities are equal in the two contexts. As this then is still just using non-contextuality (as a rationality constraint rather then an empirical notion now; to justify 'equal chances, therefore equal type B probabilities', even with non-signalling) at a later stage, and thus the non-contextuality assumption is still invoked, and the PP or born rule is still somewhat circularly presumed, albeit in a different way, at this next stage; and nothing has been achieved,
If one is attempting to justify a one shot, single case equi probability justification, in an ' a priori way', or 'rather a 'priori like wayb given the structure of the quantum laws' , then presumably the aforementioned 'necessarilly equal truth value, equal probability value' inference, is the only (non-circular) or rather appropraite kind of 'non-probabilistic' rule of inference with regard to equi-probability justification.
it would appear that the contextuality of measurement outcomes, that is invoked to give a derivation of non-trivial (non zero-one) probabilities/ born rule justification, (in some cases) simultaneously, undermines any-non-circular justification of those values in terms of some non-probabilistic rule of inference that is tacit or implicitly dependent on (in some way)'a equal evidence/equal norm squared amplitudes/chances, therefore equal probabilities' principle, as opposed to the more robust (equal truth values, which is hard to argue with
So even allowing non-signalling; so as to justify equal norm squared amplitude or chances (probability interpretation) values being invariant, between both contexts, still requires one to justify the link between objective rational credence/probability and said amplitudes squared/chances/probability interpretation values in these two contexts, it appears . To say that the these type (B) probabilities could differ between contexts, does not deny 'non signalling' ; as it does not indicate there being an physical or even psychological change at all, or even that the evidence has, or has been to believed to have changed; the best interpretation of this scenario, would these current circumstances, would be that there isn't a one true type B probability if one's type (A) probability cannot justify their values. In other words, its not the case that some some other type (A) probability is justified or is true, on account of different evidence, or some other value or type of kind of type B probability being the case in these circumstances/context; the situations are qualitatively equal lets say, but the evidence or chance, is not equal in the two contexts. This is not because the evidence/chance has 'changed' but because there never was a chance to begin with, and thus t'here exists', no chance in both contexts such that it is equal.. Likewise, there exists no chance, or any evidence in one context, such that it differs from an existent chance or piece of evidence in the other; and thus the situations are the same (if there were a chance, it would be equal, but that is part of what is needed to proven, contingent on said chances being able to derive the PP or born rule, in order to deserve the name).
Likewise one could argue that the type A probabilities may not be equal, not because they differ, but in neither context, does there exist a type A probability; and so there does not exist a type A probability in both contexts such that it is value is same; presumably it has to be the case to have a value; if there is not a value, there is no value such that its the same. Yet there is no violation of no- signalling, as there likewise exists no type A probability in context one such that its values differs, from an existent type a probability in context 2 either!
Unless I am mistaken here and there is a three part distinction
To be continued
It seems to me that you rely on hidden variables, which are out of favor to make the proof in non signaling entangled pairs, because any type of measurement in the pair causes signaling to occur. The conventional opinion is that the hidden variable does not become real until it is measured, but is preceded by a set of potentials, one of which is selected in the measurement. I won't disqualify such methods completely because of hidden momentum demonstrated by Feynman, but will suggest anything pre-existing will be altered by the measurement.
A lot of work on other proofs are regarded as inconclusive or circular logic. There remains the chance someone will find a new method that meets your criteria, even if the method arises from the uncertainty and indeterminacy of the system. The possibility cannot be disproved exactly.
Dear Jerry,
It depends on what they are trying to establish through non-signalling; even Barnum who was one of the first ones to mention with regard to a justification of the assumptions in zureks derivation, did appear to say that it still appeals to non-contextuality and thus another invariance assumption in another way.
Even if they could establish non-signalling it would not really establish much; is it being to
justify
1.is it equal truth values?
2.equal chances?, (propensities, subjective/objective (bayesian) degrees of belief interpretation, subjective betting odds interpretation), relative frequencies infinite or finite, the classico-logical interpretation or something other thing; the amplitude squared interpretation, the measure interpretation) but these are not the same thing as the true logical or rational objective probability/degrees of belief, or the one true objective, come what may betting odds, which is what we are trying to prove in (4)
3.equal amplitudes squared?,
4.equal logical, rational degrees of belief (the latter being the rhs of the PP or born rule?.
(3)'s already a given, and likewise i would say that (2) is already a given, so either no-signalling is either a justification of what is already given, or they justifying (4) via (2) or (3) which is just saying that equal amplitudes/equal chances equal '(4) style' (logical/rational objective degree of belief) probabilities, which appeals again to non-contextuality as a rationality assumption (which is what it is, in my view) and still presumes what they are aiming to prove, and presume the assumption (non-contextuality) they are aiming to justify. UNless they are going by route (1)
Its already given that the amplitudes mod squared are the same, so non-contextuality is a rationality assumption which takes one from equal mod squared amplitudes to equal logical probabiliites their equal value already being given (in some sense); and likewise that the 'chances' are the same.
But then one still has to apply something like the PP, the born rule, the principle of indifference or non-contextuality to justify that the logical probabilities, objective rational degrees of belief are the same. ie non signalling might get you that the the chances are the same, but one then has to argue that the logical probabilities are the same, and that could only then be done be an implicit assumption of 'equal chances in both contexts' therefore equal logical probabilities, which just is an application or use of non-contextuality (as a rationality assumption which it always was)), or the PP, or the qualitative born rule, the very thing one was using non-signally to justify in the first class.
Unless they use it to establish (4) via (1)equal truth values, but its precisely the fact that they must have opposite truth values which is used in the second horn of the proof to provide a non probabilistic justification of equi-probability (in terms of necessarily equal truth values); where they invoke contextuality of measurement outcomes or entanglement in a singlet state (anti correlation).ie P(~A) in first context has same probability as P(A) in second context,so that by anti-corrrelation; and P(A) in first context has the same probability as P(A) in the first second context, hence P(A) and P(~A) take the same value (qualitative value at least); but this anti-correlation undercuts itself as it demands that A in both contexts have different truth values thus undermining the first horn in term of a necessary equal truth value link, on pain of contradiction.
The justification of the first horn would undercut the needed justification of the second; because then they would no longer be anti-correlated; and one could not show that P(A) and P(~A) are both equal to some( P(B) or some P(~A) in another context) virtue of having the same truth values as that B, or (~A in context 2); that is in both direction as a biconditional, for both A and ~A in context 1 ie if (if A is true, then (then the context 2 proposition;{ B } is true or in this case (or ~A in context 2) is true, and (if A is false, B (or the context 2 proposition ~A is false) and likewise for ~A in context1 .
To me ; necessarily equal truth values (even if just physically necessarily which is presumably what it generally is) the only kind of single case non-circular justification of equi-probability; at least qualitatively (if probability is measure of possibility in what sense could one say that one is more or less probable then the other when there are no possible states, let alone, the fact that there no( more or less) possible states, in which one event occurs and the other does not ;
That is one one could/should be able to in principle construct some kind of argument to show that the person is not merely irrational insofar as they are irrational but are literally contradicting themselves. By an argument of this kind of form
1. qualitative dutch-book or conditional bet qualitative dutchbook (either with single case odds or an epistemic argument), to show the odds that one accepts on bet on A and a bet on ~A must be the same but not what those odds are . Not to to establish that that they must be 1-1 ( A and ~A are equ-probable), and thus also establishing that its a probability function at the same time, but rather that if you bet 1-1 on A, or 2-1 on A or 3-1, then one must correspondingly bet 1-1 on ~A (if 1-1 on A) or 2-1 on ~A( if 2-1 is accepted on A), if they are judged equi-probable. This is is weaker than the standard dutch-book argument for probabilism and not as open to its objections.
Or
2. which amounts to nearly the same thing to show that the person (by qualitative dominance, conditional logic dominance) is not merely irrational but literally contradicting themselves. That is without using the principle of indifference, or using equal proportions of possible truth values of 1, but necessarily equal truth values (or counterfactual if the context is relevant)
This would make the justification on a similar footing of logic, ie and the incoherence, in violating it likewise; that in believing of two logical/physical equivalents, that one is the case, and the 'other is or could be' not at the same time. -
So that contextuality or entanglement used to derive the born undercuts (and thus the second horn by non-probabilistic means), it appears, undermines, its own non-probabilistic justification (ie of non-contextuality), the first horn
Dear jerry,
When you talk about non-signalling entangled pairs are you saying that hidden variables commit me, or others to signalling and violations of locality?(which is generally supposed (perhaps controversially) to cover all cases of entanglement, at least if they are space like separated and if there is any chance of relativity being respected; not saying that may or may not be proven wrong one day) Likewise when you say that the hidden variable does not become real until measurement;, do you mean until after (and/or as a result of measurement?, and that the hidden variable's temporal position or its coming into existent is also after the measurement (not just when it is determined by caused being after the measurement)?; ie its not retrocausal or changing the past, making some hidden variable pre-exist in the past, which at the time of the past, it did not exist- which is bordering, if not contradictory I think)
I think the bohmians commit to that, as despite the fact that its the 'conventional hidden variable account' , their account is not really a hidden variable (or preexistent outcome) account at all, but a hidden causal parameter state account (potentials and momentum of other particles or particles, as causes, not pre-existent variables of the measurement outcome in an earlier state, I presume they are distinct from, but which causes the measurement outcome; where these are common causes of both the the measurement and the outcome, so that the measurement itself does not actually cause the outcome or have any direct active role. Although I am not sure though, if that makes their account non-local or super-deterministic (i think nonlocal but maybe both).I presume in their account there is a distinct common causes for the (measurement and outcome) for each observable in an entangled pair but not a singular common cause for both obervable; so that their account is more non local when it comes to entanglement, that somehow the first causal.potential parameter determines the measurement settings or rather non-localy causes the potential the hidden cause of the second?
In my account the measurement does have an actual causal (or something like) role in making the outcome occurs; but not by changing what I call the quasi-state or quasi-potentials (i call them resistance) but how they act (aspectual, perspectival? perhaps- some admixture of the modal/relational view and a propensity/indeterministic stochastic mechanic that is everett like in some ways)
It commits me, in my account to a holism I think but non necessarily. In the current paper i am talking about modal correlations (contextuality correlations, or modal entanglement), not so much entanglement across space and time but will be discussed when it comes to frequencies in the next part; where there is only one quasi state for the entire entangled system
( I am not sure i would say that its even located in time and space); where different measurement setting induce different outcomes and correlations due to probability inequalites (if the less probable occurs event occurs, the more probable does, very roughly speaking), where which events are compared by this inequalities, in this way depends on the question you ask and the measurement settings). I will have to look into this more for more complicated systems so that the right statistics are produced, but it should not affect which variable is the case, or the outcome on the other pair. It does determine which concrete outcome, that same quasi state gets transformed into after measurement, where it can go either way, but it does not change the pre-existent (if you can call it) quasi state itself, as it never was a pre-existent cause or variable or eigenvalue to begin with (until after measurement), and so I would not count it as a change (as there was never an existent piece of concrete reality to change)
My account is more in general, an account of indeterministic chance (not qm specifically or as yet), which involves a quasi state which is also indeterminsitically, which in a holistic way makes a an outcome after and as a result of measurement, its not a pre-existent cause or piece of reality; not so much as a contextual hidden variable (at least not in the traditional sense; the variable acts contextually on the measurement or the question you ask even, but its existence is not contextual);What one asks or what measurement can be conducted can be indeterministic or freedom, so there is no super-determinism either;as it does not change which variable exists, but how it acts),
Even this quasi state could come after the measurement outcome; or setting (although not determined by it; how it operates is. (although it requires an odd view of modality to get the strict indicative/morgenbesser-counterfactual definiteness, 'if then relations' to even possibly (possibly) come out right under non-determinism so that qualitative (non-probabilistic dominance may be used)
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